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 PDE Seminar

For further information, or to suggest a PDE Seminar speaker, please contact:

Misha Perepelitsa Gabriela Jaramillo William Fitzgibbon


Click Here for prior year seminars

Seminar on Partial Differential Equations
Fall 2020

All talks in ZOOM, Friday at 2:00 PM
Click here for Fall 2019

September 4
September 11  
 September 18  
September 25  
October 2  
October 9  
October 16 Kris Jenssenn
Penn State

Title: Self-similar Compressible Euler Flows
In 1942, Guderley gave examples of self-similar compressible flows in which a symmetric shock wave collapses at the origin and reflects a new outgoing spherical shock. Besides their relevance for bomb-making and Inertial Confinement Fusion, these solutions are of interest in mathematical fluid dynamics. In particular, they demonstrate how wave focusing can generate unbounded solutions from bounded initial data, also in nonlinear models. We shall report on recent work showing that Guderley solutions provide genuine weak solutions of the multi-d, compressible Euler system.

On the other hand, these same solutions exhibit zero pressure regions due to vanishing temperature, and they are therefore of border-line physicality.
To address this we show that the Euler model admits other (self-similar) solutions which suffer blow up in amplitude, but with everywhere strictly positive pressure. Thus, the mechanism of wave focusing is sufficiently strong, on its own, to generate unbounded amplitudes.

Joint work with Charis Tsikkou (University of West Virginia).
October 23  
October 30  
November 6 Stephanie Dodson
Uiniversity of California, Davis

Title: Behavior of Spiral Wave Spectrum with a Rank-Deficient Diffusion Matrix

Spiral waves emerge in numerous pattern forming systems and are commonly modeled with reaction-diffusion systems. Some systems used to model biological processes, such as ion-channel models, fall under the reaction-diffusion category and have one or more non-diffusing species which results in a rank-deficient diffusion matrix. Previous research has focused on understanding the spectral stability of spiral waves, but the impact of a rank-deficient diffusion matrix on the spiral spectra are as of now unknown. In this talk, we will use a general two-variable reaction diffusion system to investigate changes in the essential and absolute spectrum of spiral waves under the cases of a diffusive and non-diffusive slow variable. The spiral spectra is found to have profound and unexpected differences for these two limits. We will present expansions for the essential spectrum curves under the two diffusion cases, and our results indicate the diffusiveless species is directly responsible for observed changes in the essential spectrum. Moreover, we predict locations for the absolute spectrum in the case of a non-diffusing slow variable. Predictions are numerically confirmed with the Barkley and Karma models.
November 13  
November 20 Jason Bramburger
University of Washington, Seattle
Title: Sharp bounds on minimum wave speeds using polynomial optimization

Many monostable reaction-diffusion equations admit one-dimensional travelling waves if and only if the wave speed is sufficiently high. The values of these minimum wave speeds are not known exactly, except in a few simple cases. In this talk I will present methods for finding upper and lower bounds on minimum wave speeds by constructing trapping boundaries for dynamical systems whose heteroclinic connections correspond to the travelling wave. When the reaction-diffusion equations being studied have polynomial nonlinearities, this approach can be implemented computationally using polynomial optimization. For scalar reaction-diffusion equations, I will present a general method and apply it to a modified FKPP equation from the literature where minimum wave speeds were previously unknown. I will also discuss extensions to multi-component reaction-diffusion equations, including a cubic autocatalysis model from the literature.
November 27  
December 4 Glenn Webb
Vanderbilt University
Title: A Mathematical Model of CT Scans for Lung Cancer Diagnosis

A diffusive logistic partial differential equation is developed
to provide a dynamic model corresponding to the histograms of Hounsfield
units obtained from the ground glass opacity measurements of lung CT
scans. The model simulations provide agreement with the five phases of
lung cancer tumor development, as pro filed by the histograms. The model
quantifies the tumor growth dynamics and can be used to schedule
additional CT scans or medical procedures. The model is compared to
patient data at Vanderbilt University Veterans Administration Hospital.

Small changes/shifts in the dates may be possible.

   University of Houston    ---    Last modified:  February 13, 2018 - 1:32 PM